Ismp 2012

A Divide-and-Bridge Heuristic for Steiner Minimal Trees on the

Euclidean Plane

Badri Toppur, Ph.D.

Associate Professor of Operational Research

Great Lakes Institute of Management Chennai, India

INTRODUCTION TO THE STEINER PROBLEM

• The Steiner problem in cost management has its origins from Fermat’s problem in the Renaissance period.

• Connect three villages in the area, using another site if necessary so that the total interconnecting length is minimum.

• Torricelli, Napolean and Steiner among others have worked on solutions that use only compass and ruler.

21st Symposium on Mathematical Programming, TU Berlin

COMPASS AND RULER CONSTRUCTIONS

• The location of the Steiner site can be found through elementary geometric operations for three fixed terminals.

• The sites are found within the convex hull of the fixed terminals.

• Triangles erected on the sides of the triangle formed by the fixed terminals.

• The circumscribing circles around the erected triangles intersect precisely at the location of the Steiner site.

• There are some alternative Euclidean constructions.• These methods can be extended to handle more sites.


SOLVING THE STEINER PROBLEM USING A COMPUTER

• The scale of Fermat’s problem to be tackled has become large because of the extreme globalization and nationalization of the supply chain.

• The large-scale problem is called the Steiner problem and can be described using algebraic notation that makes for a computer solution.

• Connect sites v1, v2, …vn in the Euclidean plane, using other sites s1,….sn-2 if necessary so that the total interconnecting length of the resulting tree is minimum.


ENUMERATION METHODS

• Melzak, Gilbert and Pollak provided the earliest enumeration approaches to this NP hard problem.

• Winter, Zachariasen and others have implemented good methods for the 2d problem.

• Warren Smith’s Branch and Bound is the best enumeration method for any dimension, though with an exponential time limitation.

• Sets with up to 15 sites can be solve exactly in reasonable time.

• A number of heuristics have been implemented to obtain good solutions for more than 15 sites, in reasonable time.


COMPUTATIONAL GEOMETRY HEURISTIC

• James MacGregor Smith in 1978, in collaboration with his guide Judith Liebman, published numerous polynomial time methods using Computational Geometry principles to solve the Steiner problem in 2d. His application was in architectural design.

• He was my advisor at Umass, Amherst when I worked in 1996 on a heuristic for the same problem but in 3d. We worked on atomic coordinate datasets of compounds such as Silk, Actin and Collagen.


GRAPH ALGORITHM HEURISTIC

• My heuristic was based on Depth First Search, a graph algorithm technique I had learnt earlier at Clemson University, during my M.S.

• I used DFS of the Minimum Spanning Tree using the optimization method for a fixed topology that is a part of the exponential algorithm design of Warren D. Smith.


DIVIDE AND BRIDGE HEURISTIC• This heuristic is inspired by three popular Computer

Science principles.

Lexicographic Sort

Divide and Conquer

Local Search for Best Topology

• The first two principles were used successfully by Preparata and Hong in their convex hull algorithm

• In addition to these two methods, Local Search is required in this heuristic


WHAT IS LEXICOGRAPHIC SORT?


Site X coordinate Y coordinate

V1 3.0 6.0

V2 1.1 3.0

V3 1.1 2.5

V4 2.0 4.0

V5 2.0 5.0

V6 1.0 2.0

Site X coordinate Y coordinate

V6 1.0 2.0

V3 1.1 2.5

V2 1.1 3.0

V4 2.0 4.0

V5 2.0 5.0

V1 3.0 6.0

BEFORE SORT AFTER SORT


Set Size Partition Set Size Partition

6 (3, 3) 16 (4, 4, 4, 4)

7 (4, 3) 17 (5, 4, 4, 4)

8 (4, 4) 18 (5, 4, 5, 4)

9 (5, 4) 19 (5, 5, 5, 4)

10 (5, 5) 20 (5, 5, 5, 5)

11 (3, 3, 5) 21 (3, 3, 5, 5, 5)

12 (3, 3, 3, 3) 22 (3, 3, 5, 3, 3, 5)

13 (4, 3, 3, 3) 23 (3, 3, 3, 3, 3, 3, 5)

14 (4, 3, 4, 3) 24 (3, 3, 3, 3, 3, 3, 3, 3)

15 (4, 4, 4, 3) 25 (4, 3, 3, 3, 3, 3, 3, 3)

THE RECURSIVE DIVISION OF A SET


BRIDGE BETWEEN TWO TREES


BRIDGE SOLUTION SUBOPTIMAL

BRIDGE ACROSSTHE DIVIDE

SET 1 SET 2

STEINERSITE

TOPOLOGY 1IS NOT OPTIMAL


Scenario Set Size Partition Scenarios Set Size Partition

1 2 (1, 1) 9 7 (2, 5)

2 3 (1, 2) 10 7 (3, 3)

3 4 (1, 3) 11 8 (3, 4)

4 5 (1, 4) 12 8 (3, 5)

5 6 (1, 5) 13 8 (4, 4)

6 4 (2, 2) 14 9 (4, 5)

7 5 (2, 3) 15 10 (5, 5)

8 6 (2, 4)

THE FIFTEEN ELEMENTARY PARTITIONS

Local search for the best topology is essential for these fifteen scenarios. The big topology is a union of these.


SCENARIO 1 SCENARIO 2

PRIMARY BRIDGE ACROSS THE DIVIDE

SET 1 SET 2

STEINERSITE

Scenario #2 (1, 2)Scenario #1 (1, 1)


SCENARIO 3a


SET 1

SET 2

STEINERSITE

Scenario #3a (1, 3) Topology a


SCENARIO 3b


SET 1

SET 2

STEINERSITE

Scenario #3b (1, 3)

Topology b


SCENARIO 3c


SET 1

SET 2

STEINERSITE

Scenario #3c (1, 3)

Topology c


SCENARIO 4a

PRIMARY BRIDGEACROSS THE DIVIDE

SET 1

SET 2STEINERSITE

Scenario #4a (1, 4)

Topology a


SCENARIO 4b


SET 1

SET 2STEINERSITE

Scenario #4b (1, 4)

Topology b


SCENARIO 5a


SET 1

SET 2

STEINERSITE

Scenario #5a (1, 5)

Topology a


SCENARIO 5b


SET 1

SET 2

STEINERSITE

Scenario #5b (1, 5)Topology b


SCENARIO 6a


SET 1 SET 2

STEINERSITE

Topology A maynot be optimal

Scenario #6 (2, 2)

Topology A


ADD A SECONDARY BRIDGE

REMOVE THIS EDGE TOBREAK THE CYCLE

TOPOLOGY BIS OPTIMAL

Scenario #6 (2, 2)

Topology B can be obtained from Topology A by adding a secondary bridge.


• Since there are at most five fixed sites in each set, with three Steiner sites the first bridge across the divide is selected in at most 8×8 ways. Iterate and find the best one.

• The secondary bridge that creates the cycle and could lead to the other topology is selected in at most 7x7 ways. Trace out the cycle and exclude the longest edge. Iterate and find the best one.

THE CHOICE OF PRIMARY BRIDGE AND SECONDARY BRIDGE

PSEUDOCODE OF THE HEURISTICStep 1. If the cardinality of set T0 > 6, sub-divide the given set T0 into two smaller sets

T1 and T2, otherwise calculate SMT using exact algorithm.If the cardinality of set T1 > 6 divide it into two sets to obtain T3 and T4.If the cardinality of set T2 > 6 divide it into two sets to obtain T5 and T6.Continue dividing until all the sites are in subsets ( Va, Vb, Vc,…., Vp ) having three, four or five vertices.

Step 2.a Calculate the MST for each subset in the list ( Va, Vb, Vc,…., Vp) using Prim's algorithm.Step 2.b Calculate the SMT for each subset in the list ( Va, Vb, Vc,…., Vp) using Smith's

exponential algorithm.

Step 3.a Identify the closest pair of sets Vi and Vj in the list (Va, Vb, Vc, ….Vp).Step 3.b Bridge the sets Vi and Vj to obtain Vq using the bridge terminals si and sj.Step 3.c Add Vq to the end of the list. Delete the sets Vi and Vj from the list.

Step 4.a The two bridge terminals si and sj become Steiner vertices of the bridged tree Vq.Step 4.b Modify the topology of the bridged tree Vq to include the two new Steiner vertices.Step 4.c Optimize the coordinates of the bridged tree Vq using the O(N) algorithm.


PSEUDOCODE OF THE HEURISTIC CONTINUED

Step 5.a Changes in the adjacencies of the bridged tree

5.a.(i) Insert a bridge from one of the n1 nodes in Vi to one of the n2 nodes in Vj

5. a.(ii) Insert a secondary bridge, not coincident with the primary bridge above, but one that forms a cycle with it and the other edges of the tree Vq. The cycle formed is broken by excluding the most expensive edge.

5.a.(iii) Optimize Steiner site coordinates with one or more split operations. Some of the Steiner sites will be coincident.

Step 5.b Including a site in a neigbhouring set.

Step 6. Repeat Step 3, Step 4 and Step 5, until there is one tree.


PERFORMANCE ON SAUSAGES

• The heuristic requires no local search for sausage shaped data and gives optimal solutions for every problem size.


• Fejes Toth: A sausage is an arrangement of simplices that is spatially most compact.

• In 2d it is a series of abutting triangles, and in 3d it is a series of abutting tetrahedrons that spiral around an asymptotic axis.

PERFORMANCE ON RANDOM DATAWITHOUT SECONDARY BRIDGE

• The heuristic without local search, gave the optimal solution on one in five occasions for five datasets containing 2d coordinates.

• For five datasets of 3d coordinate data it did not have even that 20% accuracy.

• Therefore local search is a must after the divide-and-bridge gives a feasible solution. Things left to do:

i) Try other bridges, besides minimum length bridge.ii) Use a secondary bridge after picking a primary bridge. Break any cycle formed. iii) Move a site from one set to a neighbouring set.


SMALL SETS OF SITES IN 2D


Six Sites

Optimal SMT

DABHeuristic SMT

MST

A 2.612875 2.712754 2.679682

B 2.187850 2.187850 2.353033

C 2.527078 2.527078 2.596151

D 3.062734 3.138599 3.138598

E 3.555844 3.555844 4.570067

Seven Sites

Optimal SMT

DABHeuristic SMT

MST

A 2.515521 2.746555 2.725121

B 2.837967 2.880656 2.880656

C 3.733940 4.084624 3.745920

D 3.094959 3.200600 3.163372

E 2.529469 2.529469 2.603461

Heuristic matches Optimal solution

SMALL SETS OF SITES IN 2D


Eight Sites

Optimal SMT

DABHeuristic SMT

MST

A 3.148039 3.392032 3.196633

B 3.572636 3.845596 3.778452

C 3.269300 4.342299 3.380685

D 4.115045 4.115045 4.231006

E 3.781634 4.615251 3.864216

Nine Sites

Optimal SMT

DABHeuristic SMT

MST

A 3.888391 3.911463 4.006239

B 4.149458 4.161037 4.269023

C 3.596876 3.596876 3.647144

D 3.836754 4.010826 3.963898

E 3.177895 3.243970 3.268170

Heuristic matches Optimal solution

THANKS AND REFERENCES

• James MacGregor Smith for guidance in my initial exposure to heuristics for the problem. MacGregor~Smith, J. R. Weiss, and Minoo Patel. 1995. An O(N2) Heuristic for Steiner Minimal Trees in E3. Networks, Vol. 25, pp. 273-289.

• Warren D. Smith for the exact exponential time algorithm written in C. Smith, Warren D., 1992, How to find Steiner Minimal Trees in Euclidean d-Space., Algorithmica 7, pp. 137-177.

• Preparata, F. P. and S. J. Hong, 1977, Convex Hulls of Finite Sets of Points in Two and Three Dimensions. Communications of the ACM Volume 20, Number 2.

• John Beasley for emphasizing benchmark problems. Beasley, J. E. and F. Goffinet, 1994. A Delaunay Triangulation-Based Heuristic for the Euclidean Steiner Problem Networks, Volume 24, pp. 215-224.



ANYTHING THAT HAS NOT BEEN SUFFERED AND SOLVED TO THE END WILL RETURN.

HERMANN HESSE

Ismp 2012

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